Let the vertices of a triangle ABC be (4,3) , (7,-1) , (9,3) then the triangle is :

Options

A. Scalene
B. Isosceles
C. Equilateral
D. none of (a) , (b) ,(c)

Answer

To determine the type of triangle formed by the vertices A(4, 3), B(7, -1), and C(9, 3), we will calculate the lengths of the sides AB, BC, and AC using the distance formula.

### Step 1: Calculate the length of AB
The distance formula is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
For points A(4, 3) and B(7, -1):
- \(x_1 = 4\), \(y_1 = 3\)
- \(x_2 = 7\), \(y_2 = -1\)

Substituting these values into the formula:
\[ AB = \sqrt{(7 - 4)^2 + (-1 - 3)^2} = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 2: Calculate the length of BC
For points B(7, -1) and C(9, 3):
- \(x_1 = 7\), \(y_1 = -1\)
- \(x_2 = 9\), \(y_2 = 3\)

Substituting these values into the formula:
\[ BC = \sqrt{(9 - 7)^2 + (3 - (-1))^2} = \sqrt{(2)^2 + (4)^2} = \sqrt{4 + 16} = \sqrt{20} \]

### Step 3: Calculate the length of AC
For points A(4, 3) and C(9, 3):
- \(x_1 = 4\), \(y_1 = 3\)
- \(x_2 = 9\), \(y_2 = 3\)

Substituting these values into the formula:
\[ AC = \sqrt{(9 - 4)^2 + (3 - 3)^2} = \sqrt{(5)^2 + (0)^2} = \sqrt{25} = 5 \]

### Step 4: Analyze the lengths of the sides
Now we have the lengths of the sides:
- \(AB = 5\)
- \(BC = \sqrt{20}\)
- \(AC = 5\)

### Step 5: Determine the type of triangle
- Since \(AB = AC = 5\) and \(BC = \sqrt{20}\), we see that two sides are equal.
- This means the triangle is an **isosceles triangle**.

### Conclusion
The triangle formed by the vertices A(4, 3), B(7, -1), and C(9, 3) is an **isosceles triangle**.

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